Abstract

The purpose of this paper is to investigate the practical stability problem for impulsive discrete systems with time delays. By using Lyapunov functions and the Razumikhin-type technique, some criteria which guarantee the practical stability and uniformly asymptotically practical stability of the addressed systems are provided. Finally, two examples are presented to illustrate the criteria.

1. Introduction

As we all know, in many applications, we use discrete systems rather than continuous ones as the mathematical modeling, for example, numerical analysis, control theory, population models, and computer science [1–3]. Therefore, more and more attention has been paid to the theory of discrete systems, and some results for the stability of discrete systems have been obtained over the past few years [4–8].

The theory of practical stability has developed into a branch of the theory of motion stability [9]. Its notion is very useful, since it only needs to stabilize a system into a region of phase space. Based on this method, the desired state of a system can be unstable only if it oscillates sufficiently near this state. Recently, there has been a significant development in the theory of practical stability [10–15]. Moreover, impulses and time delays exist in many processes of dynamic systems, for example, physics, chemical technology, population dynamics, and neural networks, and they may impact systems seriously [16–30]. Therefore, it is necessary and important to analyze the practical stability of impulsive discrete systems with time delays.

In [7, 8], authors have obtained some results for asymptotic stability and exponential stability of impulsive discrete systems with time delays. Unfortunately, there is almost no result concerning uniformly asymptotically practical stability of impulsive discrete systems with time delays. The purpose of this paper is to establish some criteria which guarantee uniformly asymptotically practical stability of the addressed systems by using Lyapunov functions and the Razumikhin-type technique. This work is organized as follows. In Section 2, we introduce some basic definitions and notations. In Section 3, the main results are presented. In Section 4, two examples are discussed to illustrate the results.

2. Preliminaries

Let denote the set of nonnegative real numbers, the -dimensional real space equipped with the Euclidean norm , the set of integers, and the set of positive integers. For any , , , and set . Let . Let . The norm of is defined by . The impulse times satisfy , , and .

Consider the following impulsive discrete systems with time delays:
where , , , . For each , is defined by , . For each , , , and, for any , there exists a such that implies that .

In this paper, we assume that and satisfy certain conditions such that the solution of system (1) exists on and is unique [4]. We denote by the solution of system (1) with initial value .

For convenience, we define the following classes of functions:;
;
.

In addition, we introduce some definitions as follows.

Definition 1 (see [9]). Given two constants and , . Then, the impulsive discrete system (1) with respect to is said to be practically stable, if implies , , ,uniformly practically stable if holds, for every ,asymptotically practically stable, if holds and, for any , there exists , , such that implies , , ,uniformly asymptotically practically stable if holds and the latter part of holds for a constant , , only dependent on .

3. Main Results

Theorem 2. Assume that there exist functions , , , such that(i) are given,(ii) for ,(iii);(iv)there is a function continuous and nondecreasing for and satisfying , , such that, for any solution of system (1), , , implies that
where and
(v). Then, the system (1) with respect to is uniformly asymptotic practically stable.

Proof. Let
For any , let be the solution of system (1) through , where , and . It suffices to show that
Now, we show that
If it does not hold, then there exists a , such that . Let . Since , it is clear that . Let . Thus,
Hence, we obtain
By (7), we obtain that, for any ,
Using condition (iv), the inequality holds for all . Thus,
From (8) and (10), it can be deduced that , which is a contradiction with the condition (iv) and, thus, (6) holds.Then, it follows from condition (iii) that
Next, we claim that
If this assertion is not true, then there exists a , such that . Let . Since , we have
Let . Thus,
Hence, we obtain
Considering (15), we obtain, for any ,
Using condition (iv), the inequality holds for all . Thus,
From (16) and (18), it can be deduced that , which is a contradiction with the condition (iv) and, thus, (12) holds.Then, it follows from condition (iii) that
Similarly, it can be deduced that
By simple induction, we can prove that
It follows from conditions (ii) and (v) that
This inequality implies that the system (1) with respect to is uniformly practically stable.Next, we show that the system (1) with respect to is uniformly asymptotically practically stable. For any , , there exist numbers , , such that
Let satisfy , and , , where . We will prove that
In order to do this, we first prove that there exists a , , such that
If (25) does not hold, then, for any , , .Note that, for ,
Thus,
Hence, we obtain
Thus,
which is a contradiction. Thus, there exists a , , such that (25) holds.Next, we prove that
Let , and we show that
If (31) does not hold, then there is a such that
Let . Since
we have
Let . Note
Thus,
Hence, we obtain
On the other hand, note that, for any ,
Using condition (iv), the inequality holds for all . Thus,
From (37) and (39), it can be deduced that , which is a contradiction. Thus (31) holds.Then, from condition (iv), we get
Similarly, it can be deduced that
By simple induction, one may derive that
Thus, (30) holds.Similarly, we can prove that there exists a , such that
By simple induction, we can prove, in general, that
Therefore, when choosing , we obtain
where . Therefore,
where . The proof is complete.

Remark 3. It can be found from Theorem 2 that it requires that the distance between two adjacent impulse times cannot be too long, and meanwhile the function should decrease at impulse times. We can see that impulses do contribute to the system’s practical stability behavior. In the following, another result will be presented from the impulsive perturbation point of view, which is different from Theorem 2.

Theorem 4. Assume that there exist functions , such that(i) are given,(ii) for ,(iii), where , ,(iv), , implies that
where is a solution of system (1),(v), .
Then, the system (1) with respect to is uniformly practically stable.

Proof. For any , let be the solution of system (1) through , where and . It suffices to show that
Next, we prove that
First, we show that
If it does not hold, then there exists a such that
Let . Since , it is clear that
Thus, for ,
By condition (iv), we have that
which is a contradiction. Thus, (50) holds.From (50) and condition (iii), we obtain
Next, we show that
If this assertion is not true, then there exists a such that
Let . Since , we get
Thus, for ,
By condition (iv), we have
which is a contradiction. Thus, (56) holds.Considering (30) and condition (iii), it can be deduced that
By simple induction, we have
which, together with (50) and condition (v), yields that
Therefore, from condition (ii), we have
Thus, system (1) with respect to is uniformly practically stable.The proof is complete.

Theorem 5. Assume that there exist functions , , , such that(i) are given,(ii) for ,(iii), where ,(iv), , implies that
with , , , where is a solution of system (1);(v).
Then, the system (1) with respect to is uniformly asymptotically practically stable.

Proof. For any , let be the solution of system (1) through , where , and . From Theorem 4, it is easy to see that the system (1) with respect to is uniformly practically stable. Now, we show that the system (1) with respect to is uniformly asymptotically practically stable.For any , there exists number such that
Let be the smallest positive integer such that
We will prove that there exists such that
To this end, we first prove that there exists , , such that
In fact, when , there exists a , , such that
If (70) does not hold, it is clear that, for any ,
Thus, for ,
It follows from condition (iv) that
On the other hand, since there at least exists one point which is not an impulsive point, we obtain
Thus,
which is a contradiction. Hence, when , there exists a , , such that
Then, we claim that
If (77) does not hold, there exists a such that
Let . Since
we have that
Note that ; thus for ,
It follows from condition (iv) that
which is a contradiction. Thus, (77) holds.Considering (77) and condition (iii), it can be deduced that
Similarly, we may show
By simple induction, we can prove in general that
Thus, (69) holds.Next, we prove that there exists , , , , , such that
In fact, when , there exists a , , such that
If (87) does not hold, it is clear that, for any ,
Thus, for ,
It follows from condition (iv) that
On the other hand, since there at least exists one point which is not an impulsive point, we obtain
Thus,
which is a contradiction. Hence, when , there exists a , , such that
Similarly, we can prove that (86) holds.By simple induction, we have that
Therefore, when choosing , we obtain
From condition (ii), we have that
The proof is complete.

4. Applications

The following illustrative examples will demonstrate the effectiveness of our results.

Example 6. Consider the following impulsive discrete system:
where , , are any three constants and .

Property 1. Given constants , satisfy , and there is a constant .Then, the system (97) with respect to is uniformly asymptotically practically stable if
where .

Proof. Choose , where is a solution of system (97). Let , , , , , and then
By Theorem 2, the above property can be easily derived.

Example 7. Consider the following impulsive discrete system:
where , , are three constants and , .

Property 2. Given constants , satisfy , and there is a constant .Then, the system (100) with respect to is uniformly asymptotically practically stable if
where .

Proof. Choose , where is a solution of system (100). Let , , , , , and then